Spherical Functions on Supersymmetric Spaces and Calogero-Moser-Sutherland Operators

Abstract

In this thesis, we formulate the notion of a spherical function on a supersymmetric space, based on the canonical Hopf superalgebra structure of enveloping superalgebras. We present a comprehensive framework for spherical functions within the context of symmetric pairs (𝔤, 𝔨) of Lie superalgebras. Our analysis begins by constructing an algebraic radial component map for the spherical functions. Subsequently, we focus on the supersymmetric pair (𝔤𝔩(1|2), 𝔬𝔰𝔭(1|2)). Despite the simplicity of this pair, we uncover an interesting connection to Euler zigzag numbers. Finally, we thoroughly investigate three families of supersymmetric pairs whose restricted root systems are of type 𝐴(𝑚,𝑛). For each of these cases, we derive a differential equation for spherical functions depending on a parameter 𝑘, which takes on the values 1, ½, or (𝑚-1-2𝑛)/2. In mathematical physics literature, this differential equation is known as the Calogero-Moser-Sutherland operator (CMS), and it occurs as the Hamiltonian of the quantum 𝑛-body problem in one dimension. Our main result in this context, which extends Sergeev's earlier work, is that for the three families of supersymmetric spaces described above, the CMS operator is obtained from the action of the Casimir operator.